Euler's Method for Solving Logistic Growth Model Using MATLAB

This paper presents an efficient numerical algorithm for approximate solutions of population growth models. The algorithm is based on the homotopy analysis method (HAM). Particular attention is paid throughout this paper to outline the features of the method. The HAM provides us with a simple way to adjust and control the convergence region of the series solution by introducing the auxiliary parameters. We analyze the asymptotic behavior of solutions for three types of population growth models. The Pad approximants are effectively used in the analysis to capture the essential behavior of the population of the identical individuals.

Applied Mathematics and Computation, 2005

This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve the Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations.

IOP Conference Series: Materials Science and Engineering

Predicting the future of population number is among the important factors that affect the consideration in preparing a good management for the population. This has been done by various known method, one among them is by developing a mathematical model describing the growth of the population. The model usually takes form in a differential equation or a system of differential equations, depending on the complexity of the underlying properties of the population. The most widely used growth models currently are those having a sigmoid solution of time series, including the Verhulst logistic equation and the Gompertz equation. In this paper we consider the Allee effect of the Verhulst's logistic population model. The Allee effect is a phenomenon in biology showing a high correlation between population size or density and the mean individual fitness of the population. The method used to derive the solution is the Runge-Kutta numerical scheme, since it is in general regarded as one among the good numerical scheme which is relatively easy to implement. Further exploration is done via the fuzzy theoretical approach to accommodate the impreciseness of the initial values and parameters in the model.

2011

This article examines the popular logistic model of growth from three perspectives: its sensitivity to initial conditions; its relationship to analogous difference equation models; and the formulation of stochastic models of population growth where the mean population size satisfies the logistic relationship. The results indicate that the appealing sigmoid logistic curve is sensitive to initial conditions and care must be exercised in developing difference equation models which display the same appealing long term behavior as the logistic growth curve. It is shown that although the logistic model is appealing in terms of its simplicity its realism is questionable in terms of the degree to which it reflects demographically accepted assumptions about the probabilities of individual births and deaths in the growth of a population. In particular this lack of realism has serious implications for the computer simulation of stochastic birth and death processes where the mean population siz...

IRJET, 2021

We cannot have a sustainable planet without stabilizing population. As the world's population grows, so does the need for resources such as water, land, forests, and energy. Unfortunately, other endangered plants, animals, and natural resources pay the price for all of this "increase and demand" in an increasingly volatile and dangerous climate. This demands the development of a mathematical model capable of accurately forecasting future population growth rates and population statistics. Mathematical models, as one of the languages of science, may predict the behavior of systems based on physics, chemistry, biology, and other disciplines. Certain mathematical models may be used to accurately forecast economic and social systems, including population increase. The current work focuses on population growth mathematical modelling utilizing exponential and logistic growth models, which are nothing more than differential equations that allow us to examine population size changes over time and estimate the population of a given location at a given time. The prediction is compared to the actual population of the past, based on a model that accurately forecasts population growth rates and may be used to predict future population growth rates.

Open Journal of Mathematical Sciences

This paper presents the development of a new numerical scheme for the solution of exponential growth and decay models emanated from biological sciences. The scheme has been derived via the combination of two interpolants namely, polynomial and exponential functions. The analysis of the local truncation error of the derived scheme is investigated by means of the Taylor’s series expansion. In order to test the performance of the scheme in terms of accuracy in the context of the exact solution, four biological models were solved numerically. The absolute error has been computed successfully at each mesh point of the integration interval under consideration. The numerical results generated via the scheme agree with the exact solution and with the fifth order convergence based upon the analysis carried out. Hence, the scheme is found to be of order five, accurate and is a good approach to be included in the class of linear explicit numerical methods for the solution of initial value prob...

Communication in Biomathematical Sciences

Richards model, Gompertz model, and logistic model are widely used to describe growth model of a population. The Richards growth model is a modification of the logistic growth model. In this paper, we present a new modified logistic growth model. The proposed model was derived from a modification of the classical logistic differential equation. From the solution of the differential equation, we present a new mathematical growth model so called a WEP-modified logistic growth model for describing growth function of a living organism. We also extend the proposed model into couple WEP-modified logistic growth model. We further simulated and verified the proposed model by using chicken weight data cited from the literature. It was found that the proposed model gave more accurate predicted results compared to Richard, Gompertz, and logistic model. Therefore the proposed model could be used as an alternative model to describe individual growth.

Open Journal of Modelling and Simulation, 2014

In this paper, some theoretical mathematical aspects of the known predator-prey problem are considered by relaxing the assumptions that interaction of a predation leads to little or no effect on growth of the prey population and the prey growth rate parameter is a positive valued function of time. The predator growth model is derived considering that the prey follows a known growth models viz., Logistic and Von Bertalanffy. The result shows that the predator's population growth models look to be new functions. For either models, the predator population size either converges to a finite positive limit or to 0 or diverges to +∞. It is shown algebraically and illustrated pictorially that there is a condition at which the predator-prey population models both converge to the same finite limit. Derivations and simulation studies are provided in the paper. Analysis of equilibrium points and stability is also included.

SYMPOSIUM ON BIOMATHEMATICS 2019 (SYMOMATH 2019), 2020

In many cases, the order of a differential equation is a natural number. However, in some applications, this order can be in the form of a fractional number, so that the equation is then called a fractional differential equation. In this paper, we study the numerical solution of the fractional logistic differential equation with order α, where 0 < α ≤ 1. The equation can be considered as one of the fractional Riccati differential equations. The numerical methods we use are the Adomian decomposition method (ADM) and the variational iteration method (VIM). We use the Caputo derivative to find the solution. The effect of the fractional-order into the transient solution is studied graphically to find the interpretation in the logistic population growth model.